A duality between q-multiplicities in tensor products and q-multiplicities of weights for the root systems or D

Starting from Jacobi–Trudi type determinantal expressions for the Schur functions of types B , C and D , we define a natural q-analogue of the multiplicity [ V ( λ ) : M ( μ ) ] when M ( μ ) is a tensor product of row or column shaped modules defined by μ . We prove that these q-multiplicities are equal to certain Kostka–Foulkes polynomials related to the root systems C or D. Finally we express the corresponding multiplicities in terms of Kostka numbers